In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. We assume that, for numerical reasons, one has to time-discretize the diffusion process which typically leads to filtering that is subject to discretization bias. The approach in [16] establishes that when only having access to the time-discretized diffusion it is possible to remove the discretization bias with an estimator of finite variance. We improve on the method in [16] by introducing a modified estimator based on the recent work of [17]. We show that this new estimator is unbiased and has finite variance. Moreover, we conjecture and verify in numerical simulations that substantial gains are obtained. That is, for a given mean square error (MSE) and a particular class of multi-dimensional diffusion, the cost to achieve the said MSE falls.

翻译：本文考虑对在离散时间点上规则观测的部分可测多维扩散过程进行滤波。假设出于数值计算原因，需对扩散过程进行时间离散化，这通常会导致滤波结果存在离散化偏差。文献[16]的方法证明，当仅能获取时间离散化后的扩散过程时，可通过有限方差估计量消除离散化偏差。我们基于文献[17]的最新研究，通过引入改进的估计量对文献[16]的方法进行优化。论证表明该新估计量无偏且方差有限。此外，通过数值模拟验证并推测可取得显著增益——即针对特定类别的多维扩散过程，在给定均方误差（MSE）条件下，实现该均方误差的计算代价有所降低。